The immersed analysis concept was originally proposed in the context of the finite difference method by Peskin [1]} in 1972. This immersed boundary method (IBM) and its enhancements have been employed in a wide range of applications ever since (see, e.g., Ref. [2]} for a contemporary review). The application of the immersed element concept in the finite element setting can be traced back to the work on the partition of unity method [3]}, [4]}, generally referred to as the generalized or extended finite element method (GFEM [5]}, [6]} or XFEM [7]}, [8]}), where elements are cut in order to construct enrichment functions. The concept of cutting finite elements as an unfitted meshing technique was pioneered by Hansbo [9]}. This work can be considered as the first instance of an immersed finite element method. The pace in the development and impact of immersed finite element methods increased significantly with the introduction of the finite cell method (FCM) [10]}, [11]}, [12]}, [13]}, which combines the cut element concept with higher-order basis functions, and CutFEM [14]}, [15]}, [16]}, [17]}, [18]}, which generally employs the ghost penalty to enhance numerical stability [19]}, [14]}. Besides these prominent immersed FEM techniques, other notable examples are the aggregated finite element method (AgFEM) [21]}, [21]}, [23]}, the Cartesian grid finite element method (cgFEM) [24]}, [25]}, weighted extended B-splines (WEB-splines) [26]}, [27]} and immersed B-splines (i-splines) [28]}. Discontinuous Galerkin methods can readily be used on cut meshes after aggregation of cells, since they can be posed on polytopal meshes [29]}. In recent years, the immersed analysis concept has been considered in conjunction with isogeometric analysis [30]}, [31]} (often referred to as iga-FCM [12]} or immersogeometric analysis [33]}). In this setting, immersed methods have been demonstrated to be capable of leveraging the advantageous properties of the spline basis functions used in isogeometric analysis, while enhancing the versatility of the simulation workflow for cases where boundary-fitting spline geometries are not readily available, e.g., in scan-based analyses.
